Normal Distribution
nothing is called 95% until it has been counted
Concept

Recalibration — where it appears

Replacing each forecast value by the frequency of the event among the forecasts that carried it, so the new forecasts are calibrated by construction. It removes the reliability term of a score and leaves resolution untouched, so it cannot return information a coarse or distorted forecast has already thrown away.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Six forecasters, all calibrated, not equally useful. The resolution of six forecasters that are all perfectly calibrated, each reporting the true probability of the event given a signal that carries more or less of the latent state. The largest reliability anywhere in the family is 2.0e-33, so a calibration check passes every one of them. They are not equally good: resolution runs from exactly 0.00 for the forecaster that issues the base rate every time to 0.092758 for the one that sees everything, and their Brier scores run from 0.234237 — which is the world's own uncertainty, and the score of a table of base rates — to 0.141479. Calibration is a necessary condition that a constant forecast satisfies exactly.

Calibrated and useless

Six forecasters that are calibrated to 2·10⁻³³ run from resolution exactly 0 to 0.092758, and three forecasters with reliabilities from 0 to 0.013025 have areas under the ROC curve identical to every bit a double carries. Each measure is exactly blind to what the other one sees.

calibrate · Calibration One curve, and two forecasters that are each a single point. The ROC curve of an honest forecaster whose signal has correlation 1 with the latent state — every threshold on its probability, from the bivariate normal — and two forecasters that only ever say 0 or 1. The one an absolute-error score pays for says 1 wherever the honest probability exceeds a half: it has a true-positive rate of 0.6742 and a false-positive rate of 0.1375, and its "curve" is the two straight segments through that point, with area (TPR + TNR)/2 = 0.7684. The honest curve's area is 0.8683. Thresholding at the base rate of 0.3744 instead of at a half gives the largest area any two-valued forecaster can have here, 0.7818, and it is still 0.0866 short.

The liar with two answers

The forecaster an absolute-error score pays for says only 0 or 1, and on the ROC square it is a single point: its area is (TPR + TNR)/2 = 0.7684, against the honest forecaster's 0.8683, and it falls below the honest one on 200 of 200 counted records. No relabelling of its two answers returns what it threw away — the best recovers a Brier score short of the honest one by exactly the 0.022154 of resolution lost — and below a signal correlation of 0.7332 the same score prefers saying no every time to an honest forecast.

calibrate · Calibration

Named alongside it

The objects these essays reach for when they reach for this one.

Base rateBrier scoreDiscriminationProbability forecastReliability termResolution termROC curveClimatological forecastClosed formForecast calibrationImproper scoring ruleIsotonic regression

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